Does the use of magic by Prospero relate to colonialism in The tempest?
As Allen Carey-Webb points out in his essay referenced below, Prospero can be seen as a colonial educator who has taken upon himself the mission of using indoctrination to "civilize" the original inhabitants of the territory he controls -- in Prospero's case, Caliban:
Prospero's and Miranda's intentions in educating Caliban prefigure Macaulay's 1835 Minute on Indian Education where non-European learning is derided and English is championed in order to create a useful class of natives, "Indian in blood and colour, but English in taste, in opinions, in morals, and in intellect" (Macaulay 729). Colonial schools and educational systems in the consequent centuries taught European languages, culture, and administration to non-European subjects.
In this version of the "White Man's Burden," Prospero's magic is the foundation of his power and authority. Prospero is a ruler who governs by virtue of his knowledge of the cosmos and its regularities -- a magus, as Carey-Webb calls him. Caliban recognizes the source of his oppression when he urges Stephano and Trinculo to destroy Prospero's power by burning his magic books (III, ii, 90-5). Caliban is, however, helpless before the advanced technology, or magic, that Prospero deploys, and he is only left to himself and to the enjoyment of his island after he has repented and shown signs that he sincerely accepts the wisdom of his master.